Brian McCall

Writing a definition is a common exercise during the early stages of Geometry. An excellent geometry definition will classify, quantify, and not have a counterexample. Once a term is defined, it can be used in subsequent definitions; for example, once parallel lines are defined, they can be used in the definition of a parallelogram.

In geometry, it's imperative that you can write a good definition, becauseit will help you to understand the properties of whatever it is you're talking about.

The three key components of a good definition.

The first one, it uses previouslydefined terms.So if you've already defined what parallellines are, you can use that to definea parallelogram.

Secondly, it classifies and quantifies.That is, by classifyingit, is it a polygon?Is it a line?What is it?And quantifies how many.So if you're talking about a polygon, you'regoing to want to say how many sides.

And, last, it has no counter-example.But what is a counter-example?A counter-example is something, an example,that will make a definition orconjecture incorrect.So if you can find a counter-example to yourdefinition, you haven't written a good one.

So a short example is let's say I had asquare and I said that a square is aquadrilateral.Which means that it has four sides.And I just left my definition like that.Turned it into Mr. McCall.

Well, I'm going to say a quadrilateral,well that could be a trapezoid, whereI could draw in one pairof parallel sides.It could be a kite where we have two pairsof congruent consecutive sides.I could draw in a rhombus.I could draw in a parallelogram.I could draw in lots of counter-examplesthat would make this definition nottrue or it wouldn't make it specificenough for just a square.

Let's look at two other ones.Let's say something that's not relatedto geometry, directly, a skateboard.Let's say I define a skateboard as somethingwith wheels that you ride.Well, that's not very descriptive.This is not a good definition.First and foremost because I could saythat this could be a bike, because abike is something that haswheels that you ride.

What about a good definition?A good definition for a parallelogram is aquadrilateral with two pairs of parallelcongruent sides.Notice that we're using words thatwe probably already defined.So quadrilateral, we would have defined beforewe started defining a parallelogram.Quadrilateral has four sides.Parallel lines we say never intersect.Two lines in the same planethat never intersect.And congruent means having thesame measure or same length.

Notice I was able to write this definitionof a parallelogram using three wordsthat I've already previously definedand there's no other counter-exampleI could draw or come up with that wouldmake this not apply to a parallelogram.

So keep that in mind when you're writinggood definitions and it will help youeven on your test and quizzes.